Cruising Flight Performance!Robert Stengel, Aircraft Flight Dynamics,

MAE 331, 2016

Copyright 2016 by Robert Stengel. All rights reserved. For educational use only.http://www.princeton.edu/~stengel/MAE331.html

http://www.princeton.edu/~stengel/FlightDynamics.html

•! Definitions of airspeed•! Performance parameters•! Steady cruising flight conditions•! Breguet range equations•! Optimize cruising flight for

minimum thrust and power•! Flight envelope

Learning Objectives

Reading:!Flight Dynamics !

Aerodynamic Coefficients, 118-130!

1

Review Questions!!! What is “static margin”?!!! Is the airplane’s pitching moment sensitivity to angle of

attack linear?!!! What factors are most important in defining the

airplane’s pitching moment sensitivity to angle of attack ?!

!! Which is more important: stability or control?!!! Is the airplane’s yawing moment sensitivity to sideslip

angle linear?!!! What effect does the wing dihedral angle have on

airplane stability?!!! Why would an airplane have a “twin tail”?!!! What are “ventral fins”, and why do airplanes have/not

have them?! 2

U.S. Standard Atmosphere, 1976

http://en.wikipedia.org/wiki/U.S._Standard_Atmosphere 3

Dynamic Pressure and Mach Number! = airdensity, functionof height= !sealevele

"#h

a = speed of sound= linear functionof height

Dynamic pressure = q ! !V 2 2Mach number =V a

4

Definitions of Airspeed

•! Indicated Airspeed (IAS)

•! Calibrated Airspeed (CAS)*

Airspeed is speed of aircraft measured with respect to air massAirspeed = Inertial speed if wind speed = 0

IAS = 2 pstagnation ! pambient( ) "SL =2 ptotal ! pstatic( )

"SL

!2qc"SL

, with qc ! impact pressure

CAS = IAS corrected for instrument and position errors

=2 qc( )corr #1

!SL

* Kayton & Fried, 1969; NASA TN-D-822, 1961 5

Definitions of Airspeed

True Airspeed (TAS)*

Equivalent Airspeed (EAS)*

Airspeed is speed of aircraft measured with respect to air massAirspeed = Inertial speed if wind speed = 0

EAS = CAS corrected for compressibility effects =2 qc( )corr #2

!SL

V ! TAS = EAS !SL

!(z)= IAScorrected

!SL!(z)

Mach number

M =TASa

* Kayton & Fried, 1969; NASA TN-D-822, 19616

Checklist!"!IAS?!"!CAS?!"!EAS?!"!TAS?!"!M?!

7

Flight in the Vertical Plane!

8

Longitudinal Variables

9

Longitudinal Point-Mass Equations of Motion

!V =CT cos! "CD( ) 1

2#V 2S "mg sin$

m%CT "CD( ) 1

2#V 2S "mg sin$

m

!$ =CT sin! +CL( ) 1

2#V 2S "mgcos$

mV%CL12#V 2S "mgcos$

mV!h = " !z = "vz =V sin$!r = !x = vx =V cos$

V = velocity = Earth-relative airspeed = True airspeed with zero wind

! = flight path angleh = height (altitude)r = range

•! Assume thrust is aligned with the velocity vector (small-angle approximation for !!)

•! Mass = constant

10

Conditions for Steady, Level Flight

0 =CT ! CD( ) 1

2"V 2S

m

0 =CL12"V 2S ! mg

mV!h = 0!r = V

•! Flight path angle = 0•! Altitude = constant•! Airspeed = constant•! Dynamic pressure = constant

•! Thrust = Drag

•! Lift = Weight

11

Power and ThrustPropeller

Turbojet

Power = P = T !V = CT12"V 3S ! independent of airspeed

Thrust = T = CT12!V 2S ! independent of airspeed

Throttle Effect

T = Tmax!T = CTmax!TqS, 0 " !T " 1

12

Typical Effects of Altitude and Velocity on Power and Thrust

•! Propeller

•! Turbojet

13

Models for Altitude Effect on Turbofan Thrust

From Flight Dynamics, pp.117-118

Thrust = CT V ,!T( ) 12" h( )V 2S

= ko + k1V#( ) 1

2" h( )V 2S!T , N

ko = Static thrust coefficient at sea levelk1 = Velocity sensitivity of thrust coefficient

! = Exponent of velocity sensitivity= "2 for turbojet

#T = Throttle setting, 0,1( )$ h( ) = $SLe

"%h , $SL = 1.225 kg /m3, % = 1/ 9,042( )m"1 14

Models for Altitude Effect on Turbofan Thrust

From AeroModelMach.m in FLIGHT.m, Flight Dynamics, http://www.princeton.edu/~stengel/AeroModelMach.m

Atmos(-x(6)) : 1976 U.S. Standard Atmosphere function-x(6)= h = Altitude, m

airDens = ! = Air density at altitude h, kg/m3

u(4)= "T = Throttle setting, 0,1( )

[airDens,airPres,temp,soundSpeed] = Atmos(-x(6));Thrust = u(4) * StaticThrust * (airDens / 1.225)^0.7 * (1 - exp((-x(6) – 17000)/2000));

(airDens / 1.225)^0.7 * (1 - exp((-x(6) – 17000)/2000))

Empirical fit to match known characteristics of powerplant for generic business jet

15

Thrust of a Propeller-Driven

Aircraft

T = !P!I

PengineV

= !net

PengineV

Efficiencies decrease with airspeedEngine power decreases with altitude

Proportional to air density, w/o supercharger

With constant rpm, variable-pitch propellerwhere!P = propeller efficiency!I = ideal propulsive efficiency!netmax

" 0.85 # 0.9

16

Reciprocating-Engine Power and Specific Fuel Consumption (SFC)

•! Engine power decreases with altitude–! Proportional to air density, w/o supercharger–! Supercharger increases inlet manifold pressure,

increasing power and extending maximum altitude

17

P h( )PSL

= 1.132 ! h( )!SL

" 0.132

Anderson (Torenbeek)

SFC ! Independent of Altitude

Advance Ratio

J = VnD

from McCormick

Propeller Efficiency, ""P, and Advance Ratio, J

Effect of propeller-blade pitch angle

whereV = airspeed, m / sn = rotation rate, revolutions / sD = propeller diameter, m

18

Thrust of a Turbojet Engine

T = !mV !o!o "1#

$%

&

'(

!t!t "1#

$%

&

'( ) c "1( )+ !t

!o) c

*

+,

-

./

1/2

"1012

32

452

62

Little change in thrust with airspeed below McritDecrease with increasing altitude

!m = !mair + !mfuel

!o = pstag pambient( )(" #1)/" ; " = ratio of specific heats $1.4

!t = turbine inlet temp. freestream ambient temp.( )% c = compressor outlet temp. compressor inlet temp.( )

from Kerrebrock

19

Performance Parameters

Lift-to-Drag Ratio

Load Factor

LD = CL

CD

n = LW = L mg ,"g"s

Thrust-to-Weight Ratio TW = T mg ,"g"s

Wing Loading WS , N m2 or lb ft 2

20

Checklist!"!Flight variables?!"!Propeller vs. jet propulsion?!"!Variation with airspeed and altitude?!"!Advance ratio?!"!Wing loading?!

21

!!iiss""rriiccaall FFaacc""iidd•! Aircraft Flight Distance Records

http://en.wikipedia.org/wiki/Flight_distance_record

http://en.wikipedia.org/wiki/Flight_endurance_record•! Aircraft Flight Endurance Records

Rutan/Scaled Composites Voyager

Rutan/Virgin Atlantic Global Flyer

22

Steady, Level Flight!

23

Trimmed Lift Coefficient, CL

•! Trimmed lift coefficient, CL–! Proportional to weight and wing loading factor–! Decreases with V2

–! At constant true airspeed, increases with altitude

W = CLtrim

12!V 2"

#$%&' S = CLtrim

qS

CLtrim= 1qW S( ) = 2

!V 2 W S( ) = 2 e"h

!0V2

#$%

&'(W S( )

24

! = 1/ 9,042 m, inverse scale height of air density

Trimmed Angle of Attack, !!•! Trimmed angle of attack, !!

–! Constant if dynamic pressure and weight are constant

–! If dynamic pressure decreases, angle of attack must increase

! trim =2W "V 2S #CLo

CL!

=

1qW S( )#CLo

CL!

25

Thrust Required for Steady, Level Flight

26

Thrust Required for Steady, Level FlightTrimmed thrust

Ttrim = Dcruise =CDo

12!V 2S

"

#$

%

&'+(

2W 2

!V 2S

Minimum required thrust conditions

!Ttrim!V

= CDo"VS( )# 4$W

2

"V 3S= 0

Necessary Condition: Slope = 0

Parasitic Drag Induced Drag

27

Necessary and Sufficient Conditions for Minimum

Required Thrust

CDo!VS( ) = 4"W

2

!V 3S

Necessary Condition = Zero Slope

Sufficient Condition for a Minimum = Positive Curvature when slope = 0

! 2Ttrim!V 2 = CDo

"S( ) + 12#W2

"V 4S> 0

(+) (+) 28

Airspeed for Minimum Thrust in Steady, Level Flight

Fourth-order equation for velocityChoose the positive root

VMT =2!WS

"

#$

%

&'

(CDo

Satisfy necessary condition

V 4 = 4!CDo

" 2

#

$%&

'(W S( )2

29

Lift, Drag, and Thrust Coefficients in Minimum-Thrust Cruising Flight

CLMT= 2!VMT

2WS

"#$

%&'

=CDo

(= CL( ) L/D( )max

Lift coefficient

CDMT= CDo

+ !CLMT2 = CDo

+ !CDo

!= 2CDo

" CTMT

Drag and thrust coefficients

30

Achievable Airspeeds in Constant-Altitude Flight

•! Two equilibrium airspeeds for a given thrust or power setting–! Low speed, high CL, high !!–! High speed, low CL, low !!

•! Achievable airspeeds between minimum and maximum values with maximum thrust or power

Back Side of the Thrust Curve

31

Power Required for Steady, Level Flight

32

P = T x V

Power Required for Steady, Level Flight

Trimmed power

Ptrim =TtrimV = DcruiseV = CDo

12!V 2S

"

#$

%

&'+2(W 2

!V 2S)

*+

,

-.V

Minimum required power conditions

!Ptrim!V

= CDo

32"V 2S( ) # 2$W

2

"V 2S= 0

Parasitic Drag Induced Drag

33

Airspeed for Minimum Power in Steady,

Level Flight

•! Fourth-order equation for velocity–! Choose the positive root

VMP =2!WS

"

#$

%

&'

(3CDo

•! Satisfy necessary condition CDo

32

!V 2S( ) = 2"W2

!V 2S

•! Corresponding lift and drag coefficients CLMP

=3CDo

!CDMP

= 4CDo 34

Achievable Airspeeds for Jet in Cruising Flight

Tavail = CDqS = CDo

12!V 2S"

#$%&' +

2(W 2

!V 2S

Thrust = constant

CDo

12!V 4S"

#$%&' (TavailV

2 + 2)W2

!S= 0

V 4 ! 2TavailCDo

"SV 2 + 4#W 2

CDo"S( )2

= 0

4th-order algebraic equation for V35

Achievable Airspeeds for Jet in Cruising Flight

Solutions for V2 can be put in quadratic form and solved easily

V2 ! x; V = ± x

V 4 ! 2TavailCDo

"SV 2 + 4#W 2

CDo"S( )2

= 0

x2 + bx + c = 0

x = ! b2± b

2"#$

%&'2

! c =V 2

36

Available thrust decreases with altitudeStall limitation at low speed

Mach number effect on lift and drag increases thrust required at high speed

Thrust Required and Thrust Available for a Typical Bizjet

Tmax (h) = Tmax (SL)! SL( )e"#h!(SL)

$

%&

'

()

x

= Tmax (SL) e"#h$% '(

x= Tmax (SL)e

" x#h

Empirical correction to force thrust to zero at a given altitude, hmax. c is a convergence factor.

Tmax (h) = Tmax (SL)e! x"h 1! e! h!hmax( ) c#$ %&

37

Typical Simplified Jet Thrust Model

Thrust Required and Thrust Available for a Typical Bizjet

Typical StallLimit

38

Checklist!"!Thrust required vs. airspeed and altitude?!"!Minimum-thrust cruise vs. minimum-power

cruise?!"!Power/thrust available for climb?!"!Mach effect?!

39

Flying Qualities Becomes a Science!

Chapter 3, Airplane Stability and Control, Abzug and Larrabee!

•! What are the principal subject and scope of the chapter?!

•! What technical ideas are needed to understand the chapter?!

•! During what time period did the events covered in the chapter take place?!

•! What are the three main "takeaway" points or conclusions from the reading?!

•! What are the three most surprising or remarkable facts that you found in the reading?!

40

The Flight Envelope!

41

Flight Envelope Determined by Available Thrust

•! Flight ceiling defined by available climb rate

–! Absolute: 0 ft/min–! Service: 100 ft/min–! Performance: 200 ft/min

Excess thrust provides the ability to accelerate or

climb

•! All altitudes and airspeeds at which an aircraft can fly –! in steady, level flight –! at fixed weight

42

Additional Factors Define the Flight Envelope

!! Maximum Mach number!! Maximum allowable

aerodynamic heating!! Maximum thrust!! Maximum dynamic

pressure!! Performance ceiling!! Wing stall!! Flow-separation buffet

!! Angle of attack!! Local shock waves

Piper Dakota Stall Buffethttp://www.youtube.com/watch?v=mCCjGAtbZ4g

43

Boeing 787 Flight Envelope (HW #5, 2008)

Best Cruise Region

44

Lockheed U-2 “Coffin Corner”

45

Stall buffeting and Mach buffeting are limiting factors

Narrow corridor for safe flight

Climb Schedule

Air Commerce Act of 1926•! Airlines formed to carry mail and passengers:

–! Northwest (1926)–! Eastern (1927), bankruptcy–! Pan Am (1927), bankruptcy–! Boeing Air Transport (1927), became United (1931)–! Delta (1928), consolidated with Northwest, 2010–! American (1930)–! TWA (1930), acquired by American–! Continental (1934), consolidated with United, 2010

Boeing 40

Ford Tri-Motor Lockheed Vega

http://www.youtube.com/watch?v=3a8G87qnZz4

!!iiss""rriiccaall FFaacc""iiddss

46

Commercial Aircraft of the 1930sStreamlining, engine cowlings

Lockheed 14 Super Electra, Boeing 247

47

Douglas DC-1, DC-2, DC-3

Comfort and Elegance by the End of the Decade

Boeing 307, 1st pressurized cabin (1936), flight engineer, B-17 pre-cursor, large dorsal fin (exterior and interior)

Sleeping bunks on transcontinental planes (e.g., DC-3)Full-size dining rooms on flying boats

48

Seaplanes Became the First TransOceanic Air Transports

•! PanAm led the way–! 1st scheduled TransPacific flights(1935)–! 1st scheduled TransAtlantic flights(1938)–! 1st scheduled non-stop Trans-Atlantic flights (VS-44, 1939)

•! Boeing B-314, Vought-Sikorsky VS-44, Shorts Solent•! Superseded by more efficient landplanes (lighter, less drag)

http://www.youtube.com/watch?v=x8SkeE1h_-A

49

Checklist!"!Flight envelope?!

50

Optimal Cruising Flight!

51

Maximum Lift-to-Drag Ratio

CL( )L /Dmax =CDo

!= CLMT

LD = CL

CD=

CL

CDo+ !CL

2

! CLCD

( )!CL

=1

CDo+ "CL

2 #2"CL

2

CDo+ "CL

2( )2= 0

Satisfy necessary condition for a maximum

Lift-to-drag ratio

Lift coefficient for maximum L/D and minimum thrust are the same

52

Airspeed, Drag Coefficient, and Lift-to-Drag Ratio for L/Dmax

VL/Dmax =VMT =2!

WS

"#$

%&'

(CDo

CD( )L /Dmax = CDo+ CDo

= 2CDo

L / D( )max =CDo

!

2CDo

=1

2 !CDo

Maximum L/D depends only on induced drag factor and zero-lift drag coefficient

Induced drag factor and zero-lift drag coefficient are functions of Mach number

Airspeed

Drag Coefficient

Maximum L/D

53

Cruising Range and Specific Fuel Consumption

0 = CT !CD( ) 12"V 2S m

0 = CL12"V 2S !mg#

$%&'( mV

•! Thrust = Drag

•! Lift = Weight

•! Thrust specific fuel consumption, TSFC = cT•! Fuel mass burned per sec per unit of thrust

!mf = !cTTcT :kg skN

54

!h = 0!r =V

•! Level flight

•! Power specific fuel consumption, PSFC = cP•! Fuel mass burned per sec per unit of power

!mf = !cPPcP :kg skW

Breguet Range Equation for Jet Aircraft

drdm

= dr dtdm dt

=!r!m= V

!cTT( ) = ! VcTD

= ! LD

"#$

%&'

VcTmg

Rate of change of range with respect to weight of fuel burned

Range traveled

Range = R = dr0

R

! = " LD

#$%

&'(

VcT g

#$%

&'(Wi

Wf

!dmm

Louis Breguet, 1880-1955

dr = ! LD

"#$

%&'

VcTmg

dm

55

Maximum Range of a Jet Aircraft Flying at

Constant Altitude

Range is maximized when

Range = ! CL

CD

"#$

%&'

1cT g

"#$

%&'

2CL(SWi

Wf

)dmm1 2

=CL

CD

"

#$%

&'2cT g

"#$

%&'

2(S

mi1 2 !mf

1 2( )

Vcruise t( ) =2W t( )

CL! hfixed( )S

At constant altitude and SFC

CL

CD

!

"#$

%&= maximum

B-727

56

Breguet Range Equation for Jet Aircraft

For constant true airspeed, V = Vcruise, and SFC

R = ! LD

"#$

%&'VcruisecT g

"#$

%&'ln m( ) mi

m f

= LD

"#$

%&'VcruisecT g

"#$

%&'ln mi

mf

"

#$%

&'

= VcruiseCL

CD

!"#

$%&

1cT g

!"#

$%&ln mi

mf

!

"#$

%&

!! Vcruise(CL/CD) as large as possible!! Respect Mcrit!! ## as small as possible!! h as high as possible

MD-83

57

Maximize Jet Aircraft Range Using Optimal Cruise-Climb

Vcruise = 2W CL!S

!R!CL

"! Vcruise

CLCD

( )!CL

=

! VcruiseCL

CDo+ #CL

2( )$

%&&

'

())

!CL

= 0

Assume 2W t( ) ! h( )S = constanti.e., airplane climbs at constant TAS as fuel is burned

58

Maximize Jet Aircraft Range Using Optimal Cruise-Climb

! Vcruise CL CDo+ "CL

2( )#$ %&!CL

= 2W'S

! CL1/2 CDo

+ "CL2( )#$ %&

!CL

= 0

Optimal values: (see Supplemental Material)

CLMR=

CDo

3!: Lift Coefficient for Maximum Range

CDMR= CDo

+CDo

3= 4

3CDo

Vcruise!climb = 2W t( ) CLMR" h( )S = a h( )Mcruise-climb

a h( ) : Speed of sound; Mcruise-climb : Mach number 59

Step-Climb Approximates Optimal Cruise-Climb

!! Cruise-climb usually violates air traffic control rules!! Constant-altitude cruise does not!! Compromise: Step climb from one allowed altitude

to the next as fuel is burned

60

!!iiss""rriiccaall FFaacc""iidd

Breguet 890 Mercure

Breguet Atlantique

•! Louis Breguet (1880-1955), aviation pioneer•! Gyroplane (1905), flew vertically in 1907•! Breguet Type 1 (1909), fixed-wing aircraft•! Formed Compagnie des messageries

aériennes (1919), predecessor of Air France•! Breguet Aviation manufactured numerous

military and commericial aircraft until after World War II; teamed with BAC in SEPECAT

•! Merged with Dassault in 1971

Breguet 14

SEPECAT Jaguar

61

Checklist!"!Specific fuel consumption?!"!Breguet equation?!

"! Constant altitude?!"! Cruise-climb?!

62

Next Time:!Gliding, Climbing, and

Turning Flight!Reading:!

Flight Dynamics !Aerodynamic Coefficients, 130-141, 147-155!

63

Conditions for gliding flightParameters for maximizing climb angle and rate

Review the V-n diagramEnergy height and specific excess power

Alternative expressions for steady turning flightThe Herbst maneuver

Learning Objectives

##uupppplleemmeennttaall MMaa$$rriiaall

64

Achievable Airspeeds in Propeller-Driven

Cruising Flight

Pavail = TavailV

V 4 ! PavailVCDo

"S+ 4#W 2

CDo"S( )2

= 0

Power = constant

Solutions for V cannot be put in quadratic form; solution is more difficult, e.g., Ferrari s method

aV 4 + 0( )V 3 + 0( )V 2 + dV + e = 0

Best bet: roots in MATLAB

Back Side of the Power

Curve

65

P-51 Mustang Minimum-Thrust

Example

VMT =2!WS

"

#$

%

&'

(CDo

=2!1555.7( ) 0.947

0.0163=76.49!

m / s

Wing Span = 37 ft (9.83m)Wing Area = 235 ft 2 (21.83m2 )

Loaded Weight = 9,200 lb (3,465 kg)CDo

= 0.0163! = 0.0576

W / S = 39.3 lb / ft 2 (1555.7 N /m2 )

Altitude, mAir Density, kg/m^3 VMT, m/s

0 1.23 69.112,500 0.96 78.205,000 0.74 89.1510,000 0.41 118.87

Airspeed for minimum thrust

66

P-51 Mustang Maximum L/D

Example

VL /Dmax = VMT =76.49!

m / s

Wing Span = 37 ft (9.83m)Wing Area = 235 ft (21.83m2 )

Loaded Weight = 9,200 lb (3,465 kg)CDo

= 0.0163! = 0.0576

W / S = 1555.7 N /m2

CL( )L /Dmax =CDo

!= CLMT

= 0.531

CD( )L /Dmax = 2CDo= 0.0326

L / D( )max =1

2 !CDo

= 16.31

Altitude, mAir Density, kg/m^3 VMT, m/s

0 1.23 69.112,500 0.96 78.205,000 0.74 89.1510,000 0.41 118.87 67

Breguet Range Equation for Propeller-Driven Aircraft

drdw

=!r!w=

V!cPP( )

= !V

cPTV= !

VcPDV

= !LD"

#$

%

&'1

cPW

Rate of change of range with respect to weight of fuel burned

Range traveled

Range = R = dr0

R

! = "LD#

$%

&

'(1cP

#

$%

&

'(

Wi

Wf

! dww

Breguet 890 Mercure

68

Breguet Range Equation for Propeller-Driven Aircraft

For constant true airspeed, V = Vcruise

R = ! LD"

#$

%

&'1cP

"

#$

%

&'ln w( )Wi

Wf

=CL

CD

"

#$

%

&'1cP

"

#$

%

&'ln

Wi

Wf

"

#$$

%

&''

Range is maximized when

CL

CD

!

"#

$

%&= maximum= L

D( )max

Breguet Atlantique

69

P-51 Mustang Maximum Range (Internal Tanks only)

W = CLtrimqS

CLtrim= 1qW S( )

= 2!V 2 W S( ) = 2 e"h

!0V2

#$%

&'(W S( )

R = CL

CD

!

"#

$

%&max

1cP

!

"#

$

%&ln

Wi

Wf

!

"##

$

%&&

= 16.31( ) 10.0017!

"#

$

%&ln

3,465+6003,465

!

"#

$

%&

=1,530 km (825 nm( )

70

Maximize Jet Aircraft Range Using Optimal Cruise-Climb

! Vcruise CL CDo+ "CL

2( )#$ %&!CL

= 2w'S

! CL1/2 CDo

+ "CL2( )#$ %&

!CL

= 0

Optimal values:

CLMR=

CDo

3!: CDMR

= CDo+CDo

3= 4

3CDo

2w!S

= Constant; let CL1/2 = x, CL = x

2

""x

xCDo

+ #x4( )$

%&&

'

())=CDo

+ #x4( )* x 4#x3( )CDo

+ #x4( )2 =CDo

* 3#x4( )CDo

+ #x4( )2

71